Whenever two counties are separated by an intervening portion of a third county not exceeding one mile in width, and each of such counties has constructed or shall construct a county road to the boundary thereof, and the boards of the two counties deem it beneficial to such counties to connect the county roads by the construction and maintenance of a county road across the intervening portion of the third county, it shall be lawful for the boards of the two counties to expend jointly the county road funds of their respective counties in acquiring right-of-way for the construction, improvement, repair, and maintenance of such connecting county road and any necessary bridges thereon, in the manner provided by law for the expenditure of county road funds for the construction, improvement, repair, and maintenance of county roads lying within a county.